Transcript Number System & Codes

Number System & Codes
Number Conversions
Part II
Copyright (c) 2004
Professor Keith W. Noe
Reading Assignment
Digital Design with CPLD Applications
and VHDL, by Robert K. Dueck
Pages 6 through 17
Copyright (c) 2004
Professor Keith W. Noe
Objectives
Upon the successful completion of this lesson, you
should be able to:
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Convert decimal numbers to binary, octal, &
hexadecimal.
Convert binary numbers to octal, &
hexadecimal.
Convert octal numbers to binary.
Convert hexadecimal numbers to binary.
Copyright (c) 2004
Professor Keith W. Noe
Number Conversions
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Number conversion from base 10 to
bases 2, 8, & 16 will be discussed first.
Next conversion from binary to bases 8
and 16 will be discussed.
Then we will discuss converting base 8
to binary.
Last, we will discuss converting base 16
numbers to binary.
Copyright (c) 2004
Professor Keith W. Noe
Why so many number
systems?
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Digital & microprocessor-based electronic
circuitry use the binary number system.
Man uses the decimal number system.
Because the binary number system uses only
0 and 1, it is hard for us to work with such
huge binary numbers.
Example: 1001111101110111011011102
Errors are usually the rule when working only
in the binary number system.
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Professor Keith W. Noe
Why so many number
systems?
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Technicians function better when using
the decimal number system.
This is the number system we use every
day of our life.
The octal number system (base 8)
closely resembles the decimal number
system.
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Professor Keith W. Noe
Why so many number
systems?
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Another number system that resembles the
decimal number system is the hexadecimal
number system.
The hexadecimal or base 16 number system
has 16 symbols, some of which are the first 6
letters of the alphabet.
This number system is also used with digital
systems.
This system is usually referred to as Hex
(short for hexadecimal).
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Professor Keith W. Noe
What does this mean for me?
As a technician working on digital-based
circuits, you must be able to:
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Use all four number systems with relative
ease.
Accurately convert numbers between all four
number systems.
Computers and output circuits used by
computers output codes using one of these
four number systems.
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Professor Keith W. Noe
Number Conversions
Decimal to
Binary
Octal
Hexadecimal
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Professor Keith W. Noe
Converting Decimal Numbers
to Binary, Octal & Hex
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Converting decimal numbers to any one
of these number systems uses exactly
the same process.
The process that you will use is
repeated division with the integer
portion of the decimal number being
converted.
Copyright (c) 2004
Professor Keith W. Noe
Converting Decimal Numbers
to Binary, Octal & Hex
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Use the old style of division
95=1 R4
It is important that you use this process
as shown above.
The remainders form the number in the
new number system you are converting
the decimal number to.
Copyright (c) 2004
Professor Keith W. Noe
Number Conversion
Converting 2910 to Binary
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Professor Keith W. Noe
Decimal to Binary Conversion
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To convert a decimal number to binary,
you will repeatedly divide the decimal
number by 2 keeping track of the
remainders.
Be sure to keep track of the remainders
as the remainders are used to form the
binary equivalent number.
Copyright (c) 2004
Professor Keith W. Noe
Decimal to Binary Conversion
12= 0R1
32= 1R1
72= 3R1
14  2 = 7 R 0
29  2 = 14 R 1
To read the binary equivalent of 2910, read the
remainders from the top down: 11101
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Professor Keith W. Noe
Practice Session
Practice Converting these
decimal numbers to binary:
4410
11710
14210
25510
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Practice Session
Answers:
4410 = 1011002
11710 = 11101012
14210 = 100011102
25510 = 111111112
Copyright (c) 2004
Professor Keith W. Noe
Converting Decimal Numbers
to Octal
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Professor Keith W. Noe
Decimal to Octal Conversion
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Use the same process that you used
when converting decimal numbers to
binary.
Use the repeated division process.
When converting decimal numbers to
octal, divide the decimal number by 8.
Keep track of the remainders, The
remainders form the octal equivalent.
Copyright (c) 2004
Professor Keith W. Noe
Decimal to Octal Conversion
Convert 18310 to Octal.
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Professor Keith W. Noe
Decimal to Octal Conversion
Use Repeated Division Dividing by 8
183  8 = 22 R 7
22  8 = 2 R 6
2 8 = 0 R2
18310 = 2678
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Professor Keith W. Noe
Practice Session
Practice converting these decimal numbers to
octal.
7910
19410
20810
25510
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Professor Keith W. Noe
Practice Session
Answers
7910 = 1178
19410 = 3028
20810 = 3208
25510 = 3778
Copyright (c) 2004
Professor Keith W. Noe
Converting Decimal Numbers
to Hexadecimal
Copyright (c) 2004
Professor Keith W. Noe
Conversion of Decimal
Numbers to Hexadecimal
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The same process is used for converting
decimal numbers to hex that is for
converting decimal numbers to binary
and octal.
Use the process of repeated division
keeping track of the remainders.
When converting decimal numbers to
hex, divide the decimal number by 16.
Copyright (c) 2004
Professor Keith W. Noe
Conversion of Decimal
Numbers to Hexadecimal
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Do not forget, when remainders are 10
or higher, convert the remainder to the
appropriate letter of the alphabet.
10 = A, 11 = B, 12 = C, 13 = D, 14 =
E, and F = 15.
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Professor Keith W. Noe
Decimal-to-Hexadecimal
Conversion
Convert 19510 to Hexadecimal.
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Professor Keith W. Noe
Decimal-to-Hexadecimal
Conversion
195  16 = 12 R 3
12  16 = 0 R 12 (C)
19510 = C316
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Professor Keith W. Noe
Practice Session
Practice converting the following decimal
numbers to hexadecimal.
5710
13810
21710
25510
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Practice Session
Solutions
5710 = 3916
13810 = 8A16
21710 = D916
25510 = FF16
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Professor Keith W. Noe
Other Number System
Conversion Methods
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Professor Keith W. Noe
Other Number Conversions
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There are times when it is necessary to
convert binary numbers to octal & vice
versa;
Between the binary and hexadecimal
numbers systems.
These conversions basically do not
require math such as multiplication or
division.
Copyright (c) 2004
Professor Keith W. Noe
Binary-to-Octal Conversion
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Professor Keith W. Noe
Binary-to-Octal Conversion
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It is a simple process.
Begin by dividing the binary number
into groups of three bits each beginning
on the right.
Convert each group of three bits into its
equivalent octal number from 0 to 7.
Copyright (c) 2004
Professor Keith W. Noe
Binary-to-Octal Equivalency
000 = 0
001 = 1
010 = 2
011 = 3
100 = 4
101 = 5
110 = 6
111 = 7
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Professor Keith W. Noe
Binary-to-Octal Conversion
Convert 101110012 to Octal.
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Professor Keith W. Noe
Binary-to-Octal Conversion
10111001
10 | 111 | 001
2
7
1
101110012 = 2718
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Professor Keith W. Noe
Practice Session
Convert the following binary numbers to octal.
011011002
101100112
001010012
111111112
Copyright (c) 2004
Professor Keith W. Noe
Practice Session
Answers
011011002 = 1548
101100112 = 2638
001010012 = 0518
111111112 = 3778
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Octal-to-Binary Conversion
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Professor Keith W. Noe
Octal-to-Binary Conversion
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The method for converting octal
numbers to binary is similar to the
method used for converting binary
numbers to octal.
First, separate the octal digits.
Second, write the binary equivalent for
each of the octal digits.
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Professor Keith W. Noe
Octal-to-Binary Conversion
Convert 2538 to Binary
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Octal-to-Binary Conversion
2 5 38
2 | 5 | 3
10 101 011
2538 = 101010112
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Professor Keith W. Noe
Practice Session
Convert the following octal numbers to binary.
0378
1168
1458
2738
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Practice Session
Answers
0378 = 001111112
1168 = 010011102
1458 = 011001012
2738 = 101110112
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Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
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Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
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Converting binary numbers to
hexadecimal is similar to the process
used for converting binary numbers to
octal.
When converting binary numbers to
hexadecimal, divide the 8-bit binary
number in-half.
Copyright (c) 2004
Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
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After dividing the binary number in half,
write the hexadecimal equivalent for
each 4-bit group.
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Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
Binary to Hex
Equivalency
0000 = 0
1000 = 8
0001 = 1
1001 = 9
0010 = 2
1010 = A
0011 = 3
1011 = B
0100 = 4
1100 = C
0101 = 5
1101 = D
0110 = 6
1110 = E
0111 = 7
1111 = F
Copyright (c) 2004
Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
Convert 100111002 to Hexadecimal
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Professor Keith W. Noe
Binary-to-Hexadecimal
Conversion
10011100
1001 | 1100
9
C
100111002 = 9C16
Copyright (c) 2004
Professor Keith W. Noe
Practice Session
Convert the following binary numbers to
hexadecimal.
000010102
100111102
111100112
011111012
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Practice Session
Solutions
000010102 = 0A16
100111102 = 9E16
111100112 = F316
011111012 = 7D16
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Professor Keith W. Noe
Hexadecimal-to-Binary
Conversion
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Professor Keith W. Noe
Hexadecimal-to-Binary
Conversion
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Converting a hexadecimal number to
binary is similar to converting an octal
number to binary.
Divide the hexadecimal number into its
individual numbers (symbols).
Write the binary equivalent for each
symbol.
Copyright (c) 2004
Professor Keith W. Noe
Hexadecimal-to-Binary
Conversion
Convert 5C16 to binary.
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Professor Keith W. Noe
Hexadecimal-to-Binary
Conversion
5C
5 | C
0101 1100
5C16 = 010111002
Copyright (c) 2004
Professor Keith W. Noe
Practice Session
Convert the following hexadecimal numbers to
binary.
1B16
9416
A516
FB16
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Practice Session
Answers
1B16 = 000110112
9416 = 100101002
A516 = 101001012
FB16 = 111110112
Copyright (c) 2004
Professor Keith W. Noe
Copyright (c) 2004
Professor Keith W. Noe